Base field 6.6.300125.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 7, 2, -7, -1, 1]))
gp: K = nfinit(Polrev([-1, -2, 7, 2, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, 2, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,-17,8,22,1,-3]),K([-2,1,1,-6,-1,1]),K([-10,-29,23,44,1,-6]),K([-19,-67,49,102,3,-14]),K([-6,-20,14,29,1,-4])])
gp: E = ellinit([Polrev([-3,-17,8,22,1,-3]),Polrev([-2,1,1,-6,-1,1]),Polrev([-10,-29,23,44,1,-6]),Polrev([-19,-67,49,102,3,-14]),Polrev([-6,-20,14,29,1,-4])], K);
magma: E := EllipticCurve([K![-3,-17,8,22,1,-3],K![-2,1,1,-6,-1,1],K![-10,-29,23,44,1,-6],K![-19,-67,49,102,3,-14],K![-6,-20,14,29,1,-4]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^4+a^3+6a^2-a-4)\) | = | \((-a^4+a^3+6a^2-a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 71 \) | = | \(71\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((a^4-a^3-6a^2+a+4)\) | = | \((-a^4+a^3+6a^2-a-4)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( -71 \) | = | \(-71\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{1584785}{71} a^{5} - \frac{365527}{71} a^{4} - \frac{11832625}{71} a^{3} - \frac{4916339}{71} a^{2} + \frac{8890196}{71} a + \frac{1422813}{71} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(1\) |
Generator | $\left(-a^{5} + a^{4} + 6 a^{3} - 3 a : 2 a^{5} - 16 a^{3} - 7 a^{2} + 11 a + 3 : 1\right)$ |
Height | \(0.0030387827717437604474034092071240670275\) |
Torsion structure: | \(\Z/2\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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Torsion generator: | $\left(-3 a^{5} + a^{4} + 21 a^{3} + 9 a^{2} - 12 a - 4 : 1 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
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BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.0030387827717437604474034092071240670275 \) | ||
Period: | \( 267854.18856635194283505339506806078638 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 2.22863 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^4+a^3+6a^2-a-4)\) | \(71\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
71.6-b
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.